Spectral investigation of the noise influencing multiqubit states

Characterizing and understanding noise affecting quantum states has immense benefits in spectroscopy as well as in realizing quantum devices. Transverse relaxation times under a set of dynamical decoupling (DD) sequences with varying interpulse delays were earlier used for obtaining the noise spectral densities of single-qubit coherences. In this work, using a pair of homonuclear spins and NMR techniques, we experimentally characterize noise in certain decoherence-free subspaces. We also explore the noise of similar states in a heteronuclear spin pair. Further, using a 10-qubit system, we investigate noise profiles of various multiqubit coherences and study the scaling of noise with respect to the coherence order. Finally, using the experimentally obtained noise spectrum of the 10-qubit NOON state, we predict the performance of a Uhrig DD sequence and verify it experimentally.

Figure 1

The modulation functions $f\left(t\right)$ (left column) and the corresponding Fourier transforms, i.e., filter functions $F(\omega ,n{\tau }_f)$ (right column). The first-order sampling points of the filter functions are illustrated using a schematic spectral density function $S\left(\omega \right)$ as shown in the lowest trace of the right column.

Figure 2

Pulse sequences used to measure the noise spectrum of (a) ${\rho }_{{}_{\mathrm{LLS}}}$, (b) ${\rho }_{{}_{\mathrm{LLC}}}$, and (c) ${\rho }_{{}_{\mathrm{SL}}}$. Here, ${\tau }_1=1/\left(4J\right)$, ${\tau }_2=1/\left(4J\right)+1/\left(2\mathrm{\Delta }\nu \right)$, ${\tau }_3=1/\left(4\mathrm{\Delta }\nu \right)$, and $n$ is the number of times the loop is repeated. (d) Structure of 2,3,6-trichlorophenol. The CPMG DD sequence with spin lock along the $x$ axis is shown in the inset (DDSL).

Figure 3

(a) Experimental decay constants (dots) of 2,3,6-trichlorophenol (averaged for both protons) in a 400 MHz spectrometer for a range of $\tau$ values and for different states as indicated. The solid lines correspond to decay constants obtained from the best fit by the three-Lorentzian model as described in Eq. (10). (b) The corresponding noise spectral density bands. The dashed line at 125 Hz corresponds to the maximum harmonics sampled with $\tau =2$ ms.

Figure 4

(a) The experimental decay constants (dots) at various $\tau$ delays for the singlet state ${\rho }_{{}_{\mathrm{LLS}}}$ at two different magnetic fields, i.e., 400 and 600 MHz, as indicated. The solid lines correspond to decay constants obtained from the best fit by the three-Lorentzian model as described in Eq. (10). (b) The corresponding noise spectra.

Figure 5

(a) Pulse sequence to measure the noise spectrum in a heteronuclear spin system. (b) Molecular structure of chloroform. Here, the singlet state is prepared on ${}^1\mathrm{H}$ and ${}^{13}\mathrm{C}$ spins with a coupling constant $J_{\mathrm{CH}}=209$ Hz between them. The CPMG DD sequence is shown in the inset.

Figure 6

(a) The experimental decay constants (dots) of ${}^{13}\mathrm{C}$ chloroform at various $\tau$ delays for single-spin states ${\rho }_H=I_x$, ${\rho }_C=S_x$, and the singlet state ${\rho }_S$ at 500 MHz. The solid lines correspond to decay constants obtained from the best fit by the three-Lorentzian model as described in Eq. (10). (b) The corresponding noise spectra.

Figure 7

Pulse sequence to measure noise spectra of the MSSM states. An initial INEPT (insensitive nuclei enhanced by polarization transfer) [25] operation is used to transfer magnetization from ${}^1\mathrm{H}$ to ${}^{31}\mathrm{P}$. The PFGs $G_1$ and $G_2$ are chosen such that ${\varphi }_2\left(k\right)=-{\varphi }_1\left(k\right)$ to select out an MSSM state with a particular lopsidedness $l\left(k\right)$. A CPMG-DD sequence with composite $\pi$ pulses was used.

Figure 8

The spectral lines corresponding to various MSSM states with varying lopsidedness $l$. Each spectral line is individually normalized. The reference spectrum with all the lines is shown at the front. The structure of trimethylphosphite is also shown at the top-left-hand corner.

Figure 9

Trimethylphosphite noise spectra for various MSSM states with different lopsidedness $l$. The dashed lines parallel to the $l$ axis represent the maximum frequency (250 Hz) sampled in experiments. The inset shows the scaling of low-frequency spectral density values with $l$.

Figure 10

Decay rates vs UDD-3 cycle duration ${\tau }_c$ calculated using Eq. (7) for the experimental noise spectrum of a 10-qubit NOON state. The dots correspond to experimental results.